Carboplatin and Paclitaxel With or Without Sorafenib Tosylate in Treating Patients With Stage III or Stage IV Melanoma That Cannot Be Removed by Surgery

Inclusion Criteria:

* Histological or cytological confirmed melanoma that is metastatic or unresectable; patients must have a history of cutaneous, mucosal or unknown primary site
* Patients who have received prior systemic cytotoxic chemotherapy for treatment of melanoma are ineligible; the following groups are eligible with regard to prior systemic therapy either in the adjuvant or metastatic disease setting:

* No prior therapy
* Immunotherapy consisting of interferon, interleukin-2, granulocyte macrophage colony-stimulating factor (GM-CSF) or vaccine
* One prior investigational therapy (cannot be chemotherapy or an inhibitor of rat sarcoma \[Ras\], serine/threonine kinase \[Raf\], or mitogen-activated protein kinase kinase \[MEK\])

* NOTE: Chemotherapy given via isolated limb perfusion is allowed
* Prior radiation therapy is allowed; however, if radiation has been administered to a lesion, there must be radiographic evidence of progression of that lesion in order for that lesion to constitute measurable disease or to be included in the measured target lesions
* All sites of disease must be evaluated within 4 weeks of registration; patients must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST)
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* White blood count \>= 3,000/mm\^3
* Absolute granulocyte count \>= 1,500/mm\^3
* Platelet count \>= 100,000/mm\^3
* Serum creatinine =\< 2.0 x upper limit of normal (ULN) or serum creatinine clearance (CrCl) \>= 40 ml/min (neither drug is cleared by the kidney)
* Total bilirubin =\< 1.5 x ULN (\< 3.0 x ULN in the presence of Gilbert's disease)
* Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 x ULN (=\< 5.0 ULN in the presence of liver metastases)
* International normalized ratio (INR) =\< 1.5 and a partial thromboplastin time (PTT) within normal limits (patients who are on therapeutic anticoagulation with warfarin should have documentation of a normal prothrombin time \[PT\]/PTT prior to initiating that therapy)
* Patients must not have ocular melanoma
* Patients must have discontinued immunotherapy or radiation therapy at least 4 weeks prior to initiation of treatment and recovered from adverse events due to those agents
* Patients must not receive any other investigational agents during the period on study or the four weeks prior to initiation of treatment
* Patients must not have a history or clinical evidence of brain metastasis; patients must be evaluated with a head magnetic resonance imaging (MRI) within 4 weeks prior to enrollment
* Patients must not have other current malignancies, other than basal cell skin cancer, squamous cell skin cancer, in situ cervical cancer, ductal or lobular carcinoma in situ of the breast; patients with other malignancies are eligible if they have been continuously disease-free for \>= 5 years prior to the time of randomization
* Patients must not have any evidence of bleeding diathesis
* Patients must not have a serious intercurrent illness including, but not limited to, ongoing or active infection requiring parenteral antibiotics, clinically significant cardiovascular disease (e.g. uncontrolled hypertension, myocardial infarction, unstable angina), New York Heart Association grade II or greater congestive heart failure, serious cardiac arrhythmia requiring medication, or grade II or greater peripheral vascular disease within 1 year prior to study entry, or psychiatric illness/social situations that would limit compliance with study requirements
* Patients must not be taking cytochrome P450 enzyme-inducing antiepileptic drugs (phenytoin, carbamazepine or phenobarbital), rifampin or St. John's Wort
* Women must not be pregnant or breast-feeding
* All females of childbearing potential must have a blood test or urine study within 4 weeks prior to registration to rule out pregnancy
* Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant while participating in this study, she should inform her treating physician immediately; if a man impregnates a woman while participating in this study, he should inform his treating physician immediately as well
* Human immunodeficiency virus (HIV)-positive patients are excluded from the study